On the equation $y^m = P(x)$

Schinzel, Andrzej; Tijdeman, R.

doi:10.4064/aa-31-2-199-204

Cited by 100 publications

(77 citation statements)

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“…By using the theory of linear forms in logarithms, Schinzel and Tijdeman [18] proved that for all solutions of (13), z < c 19 where c 19 is an effectively computable number depending only on n and the height H(f) of / . For various generalizations and related results, we refer to [20] and [24].…”

Section: Super-elliptic Equationsmentioning

confidence: 99%

Bounds for the solutions of some diophantine equations in terms of discriminants

1991

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Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

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Section: Super-elliptic Equationsmentioning

confidence: 99%

Bounds for the solutions of some diophantine equations in terms of discriminants

1991

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“…In 1976, Schinzel and Tijdeman [23] proved that there is an effectively computable number C, depending only on f , such that (1.1) has no solutions x, y ∈ Z with y = 0, ±1 if m > C. The proofs of Baker and of Schinzel and Tijdeman are both based on Baker's results on linear forms in logarithms of algebraic numbers.…”

Section: Introductionmentioning

confidence: 99%

Effective results for hyper- and superelliptic equations over number fields

Bérczes

Evertse

Győry

2013

Publ. Math. Debrecen

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Abstract. Let f be a polynomial with coefficients in the ring O S of S-integers of a given number field K, b a non-zero S-integer, and m an integer ≥ 2. Suppose that f has no multiple zeros. We consider the equation (*) f (x) = by m in x, y ∈ O S . In the present paper we give explicit upper bounds in terms of K, S, b, f, m for the heights of the solutions of (*). Further, we give an explicit bound C in terms of K, S, b, f such that if m > C then (*) has only solutions with y = 0 or a root of unity. Our results are more detailed versions of work of Trelina, Brindza, and Shorey and Tijdeman. The results in the present paper are needed in a forthcoming paper of ours on Diophantine equations over integral domains which are finitely generated over Z.

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“…For (very) small values of m we will resolve (2). More precisely, we prove the following (20,8,2,9), (20,8,8,3).…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%